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Concept
Sierpiński triangle
Summary
The Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.
There are many different ways of constructing the Sierpinski triangle.
The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets:
Start with an equilateral triangle.
Subdivide it into four smaller congruent equilateral triangles and remove the central triangle.
Repeat step 2 with each of the remaining smaller triangles infinitely.
Each removed triangle (a trema) is topologically an open set.
This process of recursively removing triangles is an example of a finite subdivision rule.
The same sequence of shapes, converging to the Sierpiński triangle, can alternatively be generated by the following steps:
Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpiński triangle uses an equilateral triangle with a base parallel to the horizontal axis (first image).
Shrink the triangle to 1/2 height and 1/2 width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner (image 2). Note the emergence of the central hole—because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski's triangle.)
Repeat step 2 with each of the smaller triangles (image 3 and so on).
Note that this infinite process is not dependent upon the starting shape being a triangle—it is just clearer that way.
Official source
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